Cross section and transverse single-spin asymmetry of
η mesons in p
↑þ p
collisions at
p
ffiffi
s
¼ 200 GeV at forward rapidity
A. Adare,13C. Aidala,43,44N. N. Ajitanand,62Y. Akiba,56,57R. Akimoto,12H. Al-Bataineh,50J. Alexander,62M. Alfred,24 A. Angerami,14K. Aoki,35,56 N. Apadula,29,63Y. Aramaki,12,56H. Asano,35,56E. T. Atomssa,36,63R. Averbeck,63 T. C. Awes,52B. Azmoun,7V. Babintsev,25M. Bai,6G. Baksay,19L. Baksay,19N. S. Bandara,43B. Bannier,63K. N. Barish,8 B. Bassalleck,49A. T. Basye,1 S. Bathe,5,8,57V. Baublis,55C. Baumann,45A. Bazilevsky,7 M. Beaumier,8 S. Beckman,13 S. Belikov,7,* R. Belmont,44,67 R. Bennett,63A. Berdnikov,59Y. Berdnikov,59J. H. Bhom,71D. Black,8 D. S. Blau,34 J. S. Bok,50,71K. Boyle,57,63M. L. Brooks,39J. Bryslawskyj,5H. Buesching,7V. Bumazhnov,25G. Bunce,7,57S. Butsyk,39
S. Campbell,29,63A. Caringi,46 C.-H. Chen,57,63C. Y. Chi,14M. Chiu,7 I. J. Choi,26,71J. B. Choi,10R. K. Choudhury,4 P. Christiansen,41T. Chujo,66P. Chung,62O. Chvala,8V. Cianciolo,52Z. Citron,63,69B. A. Cole,14Z. Conesa del Valle,36
M. Connors,63M. Csanád,17T. Csörgő,70 T. Dahms,63S. Dairaku,35,56I. Danchev,67K. Das,20A. Datta,43,49 M. S. Daugherity,1 G. David,7 M. K. Dayananda,21 K. DeBlasio,49K. Dehmelt,63A. Denisov,25A. Deshpande,57,63 E. J. Desmond,7K. V. Dharmawardane,50O. Dietzsch,60L. Ding,29A. Dion,29,63J. H. Do,71M. Donadelli,60O. Drapier,36 A. Drees,63K. A. Drees,6 J. M. Durham,39,63 A. Durum,25D. Dutta,4 L. D’Orazio,42 S. Edwards,20Y. V. Efremenko,52 F. Ellinghaus,13T. Engelmore,14 A. Enokizono,52,56,58 H. En’yo,56,57S. Esumi,66K. O. Eyser,7 B. Fadem,46N. Feege,63
D. E. Fields,49 M. Finger,9 M. Finger, Jr.,9 F. Fleuret,36 S. L. Fokin,34Z. Fraenkel,69,*J. E. Frantz,51,63A. Franz,7 A. D. Frawley,20K. Fujiwara,56Y. Fukao,56T. Fusayasu,48C. Gal,63 P. Gallus,15P. Garg,3 I. Garishvili,64 H. Ge,63 F. Giordano,26A. Glenn,38H. Gong,63M. Gonin,36Y. Goto,56,57R. Granier de Cassagnac,36N. Grau,2,14S. V. Greene,67 G. Grim,39M. Grosse Perdekamp,26Y. Gu,62T. Gunji,12H. Guragain,21H.-Å. Gustafsson,41,*T. Hachiya,56J. S. Haggerty,7 K. I. Hahn,18H. Hamagaki,12J. Hamblen,64R. Han,54S. Y. Han,18J. Hanks,14,63S. Hasegawa,30E. Haslum,41R. Hayano,12
X. He,21M. Heffner,38T. K. Hemmick,63T. Hester,8 J. C. Hill,29M. Hohlmann,19R. S. Hollis,8 W. Holzmann,14 K. Homma,23B. Hong,33T. Horaguchi,23D. Hornback,64T. Hoshino,23S. Huang,67 T. Ichihara,56,57 R. Ichimiya,56
Y. Ikeda,56,66K. Imai,30,35,56 Y. Imazu,56M. Inaba,66 A. Iordanova,8 D. Isenhower,1 M. Ishihara,56M. Issah,67 D. Ivanischev,55D. Ivanishchev,55Y. Iwanaga,23B. V. Jacak,63S. J. Jeon,47M. Jezghani,21J. Jia,7,62X. Jiang,39J. Jin,14
B. M. Johnson,7 T. Jones,1E. Joo,33K. S. Joo,47 D. Jouan,53D. S. Jumper,1,26F. Kajihara,12J. Kamin,63J. H. Kang,71 J. S. Kang,22J. Kapustinsky,39K. Karatsu,35,56 M. Kasai,56,58 D. Kawall,43,57 M. Kawashima,56,58 A. V. Kazantsev,34 T. Kempel,29J. A. Key,49V. Khachatryan,63A. Khanzadeev,55K. Kihara,66K. M. Kijima,23J. Kikuchi,68 A. Kim,18 B. I. Kim,33C. Kim,33D. H. Kim,18D. J. Kim,31E.-J. Kim,10H.-J. Kim,71M. Kim,61Y.-J. Kim,26Y. K. Kim,22E. Kinney,13 Á. Kiss,17E. Kistenev,7J. Klatsky,20D. Kleinjan,8P. Kline,63T. Koblesky,13L. Kochenda,55M. Kofarago,17B. Komkov,55
M. Konno,66J. Koster,26,57D. Kotov,55,59A. Král,15A. Kravitz,14G. J. Kunde,39K. Kurita,56,58 M. Kurosawa,56,57 Y. Kwon,71G. S. Kyle,50R. Lacey,62Y. S. Lai,14J. G. Lajoie,29A. Lebedev,29D. M. Lee,39J. Lee,18K. B. Lee,33,39 K. S. Lee,33S. H. Lee,63M. J. Leitch,39M. A. L. Leite,60M. Leitgab,26X. Li,11P. Lichtenwalner,46P. Liebing,57S. H. Lim,71 L. A. Linden Levy,13T. Liška,15H. Liu,39M. X. Liu,39B. Love,67D. Lynch,7C. F. Maguire,67Y. I. Makdisi,6M. Makek,69,72 M. D. Malik,49A. Manion,63V. I. Manko,34E. Mannel,7,14Y. Mao,54,56 H. Masui,66F. Matathias,14M. McCumber,39,63
P. L. McGaughey,39D. McGlinchey,13,20C. McKinney,26 N. Means,63A. Meles,50M. Mendoza,8 B. Meredith,14,26 Y. Miake,66T. Mibe,32A. C. Mignerey,42K. Miki,56,66A. J. Miller,1 A. Milov,7,69D. K. Mishra,4 J. T. Mitchell,7 S. Miyasaka,56,65S. Mizuno,56,66 A. K. Mohanty,4P. Montuenga,26H. J. Moon,47T. Moon,71Y. Morino,12A. Morreale,8
D. P. Morrison,7,† T. V. Moukhanova,34T. Murakami,35,56 J. Murata,56,58A. Mwai,62 S. Nagamiya,32,56J. L. Nagle,13,‡ M. Naglis,69 M. I. Nagy,17,70I. Nakagawa,56,57 H. Nakagomi,56,66Y. Nakamiya,23K. R. Nakamura,35,56T. Nakamura,56
K. Nakano,56,65S. Nam,18C. Nattrass,64P. K. Netrakanti,4 J. Newby,38M. Nguyen,63M. Nihashi,23,56T. Niida,66 R. Nouicer,7,57N. Novitzky,31A. S. Nyanin,34C. Oakley,21E. O’Brien,7S. X. Oda,12C. A. Ogilvie,29M. Oka,66K. Okada,57 Y. Onuki,56J. D. Orjuela Koop,13A. Oskarsson,41M. Ouchida,23,56 H. Ozaki,66 K. Ozawa,12,32 R. Pak,7 V. Pantuev,27,63 V. Papavassiliou,50I. H. Park,18S. Park,61S. K. Park,33W. J. Park,33S. F. Pate,50L. Patel,21M. Patel,29H. Pei,29J.-C. Peng,26 H. Pereira,16D. V. Perepelitsa,7,14G. D. N. Perera,50D. Yu. Peressounko,34J. Perry,29R. Petti,63C. Pinkenburg,7R. Pinson,1
R. P. Pisani,7M. Proissl,63M. L. Purschke,7 H. Qu,21J. Rak,31 I. Ravinovich,69K. F. Read,52,64 S. Rembeczki,19 K. Reygers,45D. Reynolds,62V. Riabov,55Y. Riabov,55,59E. Richardson,42N. Riveli,51 D. Roach,67G. Roche,40
S. D. Rolnick,8 M. Rosati,29C. A. Rosen,13S. S. E. Rosendahl,41Z. Rowan,5J. G. Rubin,44 P. Ružička,28 B. Sahlmueller,45,63N. Saito,32T. Sakaguchi,7 K. Sakashita,56,65H. Sako,30V. Samsonov,55S. Sano,12,68M. Sarsour,21 S. Sato,30T. Sato,66S. Sawada,32 B. Schaefer,67B. K. Schmoll,64 K. Sedgwick,8 J. Seele,13,57R. Seidl,26,56,57 A. Sen,64 R. Seto,8P. Sett,4A. Sexton,42D. Sharma,63,69I. Shein,25T.-A. Shibata,56,65K. Shigaki,23M. Shimomura,29,66K. Shoji,35,56 P. Shukla,4A. Sickles,7C. L. Silva,29,39D. Silvermyr,52C. Silvestre,16K. S. Sim,33B. K. Singh,3C. P. Singh,3V. Singh,3
M. Slunečka,9 R. A. Soltz,38W. E. Sondheim,39S. P. Sorensen,64 I. V. Sourikova,7 P. W. Stankus,52E. Stenlund,41 M. Stepanov,43S. P. Stoll,7T. Sugitate,23A. Sukhanov,7T. Sumita,56J. Sun,63J. Sziklai,70E. M. Takagui,60A. Takahara,12
A. Taranenko,62 H. Themann,63D. Thomas,1 T. L. Thomas,49A. Timilsina,29 T. Todoroki,56,66 M. Togawa,57A. Toia,63 L. Tomášek,28M. Tomášek,15H. Torii,23,56M. Towell,1R. Towell,1R. S. Towell,1I. Tserruya,69Y. Tsuchimoto,23C. Vale,7 H. Valle,67H. W. van Hecke,39M. Vargyas,70E. Vazquez-Zambrano,14A. Veicht,26J. Velkovska,67R. Vértesi,70M. Virius,15 V. Vrba,15,28E. Vznuzdaev,55X. R. Wang,50D. Watanabe,23K. Watanabe,66Y. Watanabe,56,57Y. S. Watanabe,32F. Wei,29,50 R. Wei,62J. Wessels,45S. Whitaker,29S. N. White,7D. Winter,14S. Wolin,26C. L. Woody,7R. M. Wright,1M. Wysocki,13,52 B. Xia,51L. Xue,21S. Yalcin,63Y. L. Yamaguchi,12,56K. Yamaura,23R. Yang,26A. Yanovich,25J. Ying,21S. Yokkaichi,56,57
I. Yoon,61Z. You,54G. R. Young,52I. Younus,37,49 I. E. Yushmanov,34 W. A. Zajc,14A. Zelenski,6 and S. Zhou11 (PHENIX Collaboration)
1Abilene Christian University, Abilene, Texas 79699, USA
2
Department of Physics, Augustana College, Sioux Falls, South Dakota 57197, USA
3Department of Physics, Banaras Hindu University, Varanasi 221005, India
4
Bhabha Atomic Research Centre, Bombay 400 085, India
5Baruch College, City University of New York, New York, New York 10010, USA
6
Collider-Accelerator Department, Brookhaven National Laboratory, Upton, New York 11973-5000, USA
7Physics Department, Brookhaven National Laboratory, Upton, New York 11973-5000, USA
8
University of California–Riverside, Riverside, California 92521, USA
9Charles University, Ovocný trh 5, Praha 1, 116 36 Prague, Czech Republic
10
Chonbuk National University, Jeonju 561-756, Korea
11Science and Technology on Nuclear Data Laboratory, China Institute of Atomic Energy,
Beijing 102413, People’s Republic of China
12Center for Nuclear Study, Graduate School of Science, University of Tokyo, 7-3-1 Hongo,
Bunkyo, Tokyo 113-0033, Japan
13University of Colorado, Boulder, Colorado 80309, USA
14
Columbia University, New York, New York 10027, USA and Nevis Laboratories, Irvington, New York 10533, USA
15
Czech Technical University, Zikova 4, 166 36 Prague 6, Czech Republic
16Dapnia, CEA Saclay, F-91191 Gif-sur-Yvette, France
17
ELTE, Eötvös Loránd University, H-1117 Budapest, Pázmany Péter sétány 1/A, Hungary
18Ewha Womans University, Seoul 120-750, Korea
19
Florida Institute of Technology, Melbourne, Florida 32901, USA
20Florida State University, Tallahassee, Florida 32306, USA
21
Georgia State University, Atlanta, Georgia 30303, USA
22Hanyang University, Seoul 133-792, Korea
23
Hiroshima University, Kagamiyama, Higashi-Hiroshima 739-8526, Japan
24Department of Physics and Astronomy, Howard University, Washington, DC 20059, USA
25
IHEP Protvino, State Research Center of Russian Federation, Institute for High Energy Physics, Protvino 142281, Russia 26
University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, USA
27Institute for Nuclear Research of the Russian Academy of Sciences, prospekt 60-letiya Oktyabrya 7a,
Moscow 117312, Russia
28Institute of Physics, Academy of Sciences of the Czech Republic, Na Slovance 2,
182 21 Prague 8, Czech Republic
29Iowa State University, Ames, Iowa 50011, USA
30
Advanced Science Research Center, Japan Atomic Energy Agency, 2-4 Shirakata Shirane, Tokai-mura, Naka-gun, Ibaraki-ken 319-1195, Japan
31
Helsinki Institute of Physics and University of Jyväskylä, P.O.Box 35, FI-40014 Jyväskylä, Finland
32KEK, High Energy Accelerator Research Organization, Tsukuba, Ibaraki 305-0801, Japan
33
Korea University, Seoul, 136-701, Korea
34Russian Research Center“Kurchatov Institute,” Moscow, 123098 Russia
35
Kyoto University, Kyoto 606-8502, Japan
36Laboratoire Leprince-Ringuet, Ecole Polytechnique, CNRS-IN2P3,
Route de Saclay, F-91128 Palaiseau, France
37Physics Department, Lahore University of Management Sciences, Lahore 54792, Pakistan
38
Lawrence Livermore National Laboratory, Livermore, California 94550, USA
39Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA
40
LPC, Université Blaise Pascal, CNRS-IN2P3, Clermont-Fd, 63177 Aubiere Cedex, France
41Department of Physics, Lund University, Box 118, SE-221 00 Lund, Sweden
42
43Department of Physics, University of Massachusetts, Amherst, Massachusetts 01003-9337, USA 44
Department of Physics, University of Michigan, Ann Arbor, Michigan 48109-1040, USA
45Institut fur Kernphysik, University of Muenster, D-48149 Muenster, Germany
46
Muhlenberg College, Allentown, Pennsylvania 18104-5586, USA
47Myongji University, Yongin, Kyonggido 449-728, Korea
48
Nagasaki Institute of Applied Science, Nagasaki-shi, Nagasaki 851-0193, Japan
49University of New Mexico, Albuquerque, New Mexico 87131, USA
50
New Mexico State University, Las Cruces, New Mexico 88003, USA
51Department of Physics and Astronomy, Ohio University, Athens, Ohio 45701, USA
52
Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA
53IPN-Orsay, Universite Paris Sud, CNRS-IN2P3, BP1, F-91406 Orsay, France
54
Peking University, Beijing 100871, People’s Republic of China
55PNPI, Petersburg Nuclear Physics Institute, Gatchina, Leningrad Region 188300, Russia
56
RIKEN Nishina Center for Accelerator-Based Science, Wako, Saitama 351-0198, Japan
57RIKEN BNL Research Center, Brookhaven National Laboratory, Upton, New York 11973-5000, USA
58
Physics Department, Rikkyo University, 3-34-1 Nishi-Ikebukuro, Toshima, Tokyo 171-8501, Japan
59Saint Petersburg State Polytechnic University, Saint Petersburg, 195251 Russia
60
Universidade de São Paulo, Instituto de Física, Caixa Postal 66318, São Paulo CEP05315-970, Brazil
61Department of Physics and Astronomy, Seoul National University, Seoul 151-742, Korea
62
Chemistry Department, Stony Brook University, SUNY, Stony Brook, New York 11794-3400, USA
63Department of Physics and Astronomy, Stony Brook University, SUNY, Stony Brook,
New York 11794-3800, USA
64University of Tennessee, Knoxville, Tennessee 37996, USA
65
Department of Physics, Tokyo Institute of Technology, Oh-okayama, Meguro, Tokyo 152-8551, Japan
66Institute of Physics, University of Tsukuba, Tsukuba, Ibaraki 305, Japan
67
Vanderbilt University, Nashville, Tennessee 37235, USA
68Advanced Research Institute for Science and Engineering, Waseda University, 17 Kikui-cho,
Shinjuku-ku, Tokyo 162-0044, Japan
69Weizmann Institute, Rehovot 76100, Israel
70
Institute for Particle and Nuclear Physics, Wigner Research Centre for Physics, Hungarian Academy of Sciences (Wigner RCP, RMKI), 114, P.O. Box 49, Budapest,
H-1525 Budapest, Hungary
71Yonsei University, IPAP, Seoul 120-749, Korea
72
Faculty of Science, Department of Physics, University of Zagreb, Bijenička 32,
HR-10002 Zagreb, Croatia
(Received 19 June 2014; published 20 October 2014)
We present a measurement of the cross section and transverse single-spin asymmetry (AN) forη mesons
at large pseudorapidity from pffiffiffis¼ 200 GeV p↑þ p collisions. The measured cross section for 0.5 <
pT<5.0 GeV=c and 3.0 < jηj < 3.8 is well described by a next-to-leading-order
perturbative-quantum-chromodynamics calculation. The asymmetries AN have been measured as a function of Feynman-x (xF)
from 0.2 < jxFj < 0.7, as well as transverse momentum (pT) from 1.0 < pT<4.5 GeV=c. The
asymmetry averaged over positive xF is hANi ¼ 0.061 0.014. The results are consistent with prior
transverse single-spin measurements of forwardη and π0mesons at various energies in overlapping xF
ranges. Comparison of different particle species can help to determine the origin of the large observed
asymmetries in p↑þ p collisions.
DOI:10.1103/PhysRevD.90.072008 PACS numbers: 13.85.Ni, 13.88.+e, 14.20.Dh, 25.75.Dw
I. INTRODUCTION
Since the proton’s magnetic moment was revealed to be 2.79 times the size of the Dirac magnetic moment [1],
studying the internal structure of the proton has been a vibrant field of physics research. Early deep-inelastic electron-nucleon scattering (DIS) experiments found that leptons were elastically scattered off of partons[2–4], and further measurements have led to detailed understanding of the parton distribution functions (PDFs) that can be used to describe the collinear quark and gluon structure of the nucleon. At leading order in a perturbative quantum chromodynamics (pQCD) expansion in the strong coupling
*Deceased.
†PHENIX Collaboration Spokesperson.
‡PHENIX Collaboration Spokesperson.
αs, PDF fðxÞ represents the probability of a parton of flavor f carrying momentum fraction x of the total proton momentum. The PDFs themselves are nonperturbative and cannot be calculated directly in pQCD; they must instead be extracted from experimental measurements. From the development of QCD until the 1990s, exper-imental and theoretical studies focused on the one-dimensional momentum structure of the nucleon, in which the partons are treated as moving collinearly with the parent nucleon. Over the past two decades, a variety of theoretical and experimental tools have been developed to study other aspects of nucleon structure, including parton transverse dynamics within the nucleon. The measurement of transverse single spin asymmetries (SSAs) provides one window into dynamical spin-momentum correlations both in QCD bound states and in the process of partonic hadronization.
Leading-twist pQCD calculations predict very small transverse single spin asymmetries, less than Oð10−4Þ at high-pT (pT > few GeV=c)[5]. However, strikingly large transverse SSAs, up to ∼40%, have been measured at forward rapidity for hadrons produced from transversely polarized proton collisions (p↑þ p → h þ X), revealing significant spin-momentum correlations in the nonpertur-bative structure of the proton. These asymmetries have been observed for collision energies ranging from pffiffiffis¼ 4.9 to 500 GeV[6–16]and for hadron transverse momenta (pT) up to 7 GeV=c [16]. The persistence of transverse SSAs into kinematic regimes where pQCD is applicable offers an opportunity to describe this nonperturbative behavior in terms of well-defined functions using the framework of pQCD. At midrapidity, no significant AN has been observed [15,17].
Multiple approaches have been proposed to describe the large transverse SSAs observed in hadronic reactions. Transverse-momentum-dependent (TMD) PDFs include explicit dependence not only on the partonic collinear momentum fraction but also on the partonic transverse momentum (kT) within the nucleon. Similarly, TMD fragmentation functions (FFs) depend on both the collinear momentum fraction of the scattered parton acquired by the produced hadron as well as the transverse momentum of the hadron with respect to the direction of the scattered parton. Reactions involving scattering of a proton with its spin perpendicular to its momentum inducing the production of a hadron can provide sensitivity to both initial-state (PDF) and final-state (FF) effects.
Sivers proposed a TMD PDF[18,19]as a possible origin of the large observed transverse SSAs, corresponding to a correlation between the spin of the proton and the trans-verse momentum of the quarks. Semi-inclusive DIS experi-ments have found evidence for a nonzero Sivers TMD PDF
[20–23]. Collins alternatively proposed a TMD FF [24]
that generates transverse SSAs, corresponding to a corre-lation between the (transverse) polarization of a scattered
quark and the angular distribution of pions in the quark jet. The outgoing quarks in p↑þ p collisions will have a net transverse polarization if the transversity distribution in the proton is nonzero. Electron-positron annihilation, as well as semi-inclusive DIS measurements, has now found evidence for a nonzero Collins TMD FF as well as a nonzero transversity distribution [20,22,25–29]. All these results indicate that there are sizable spin-momentum correlation effects in QCD bound states as well as in the process of hadronization.
While these spin-momentum correlations are present in the proton and in the process of hadronization, inclusive hadron production in p↑þ p collisions cannot probe TMD PDFs and FFs directly as a function of kT. However, these asymmetries do have sensitivity to the TMD PDFs and FFs integrated over kT, and attempts to describe the data phenomenologically using the Sivers and Collins effects have been done[30–32].
Perturbative QCD calculations using collinear higher-twist quark-gluon correlations [33–37] can be performed and compared to data for inclusive SSAs in hadronic collisions. While these correlation functions do not contain direct information on the transverse momentum distribu-tions of partons, this approach has been related to kT moments of TMD PDFs and FFs such as the Sivers and Collins functions for multiparton correlations in the initial and final states, respectively [38]. Prior RHIC transverse SSA measurements for inclusive hadron production have been described relatively well by a combination of twist-3 effects in the initial and final states [39–42], but further refinement in both the theoretical calculations, for example through a better understanding of uncertainties, and in experimental measurements, for example through multi-differential measurements in more than one kinematic variable simultaneously, will be needed to test and under-stand these correlations in detail.
It has been predicted that TMD factorization may be broken when the partonic transverse momentum is explic-itly taken into account, and the partons in the two incoming protons can no longer be described by independent PDFs but instead become correlated across the two protons[43]. In this case any phenomenology used to describe the asymmetries might become more complex, depending on the size of the effects from factorization breaking. The breakdown of TMD factorization leads to the pre-diction of additional spin asymmetries in the case of hadron production in p↑þ p collisions [44], with the possible magnitude of any new asymmetries still unknown. These effects, due to color exchange, will be interesting to explore further at RHIC once phenomenological predictions become available.
This paper reports on measurements of the cross section and transverse single spin asymmetry for η mesons at forward pseudorapidity (3.0 < jηj < 3.8) from the 2008 RHIC data taking period at pffiffiffis¼ 200 GeV. A total
integrated luminosity of L ¼ 6.65 pb−1 was sampled for these results. The measurement of different produced particle species will help to advance our understanding of the transverse SSAs (AN) observed in p↑þ p collisions. The comparison of pions, η mesons, and kaons can shed light on initial- versus final-state spin-momentum correla-tions as well as possible isospin, strangeness, and mass effects.
A review of the RHIC polarized pþ p collider facility and the PHENIX experiment and detectors used for the measurements is given (Sec.II), followed by a description of the analysis procedure (Sec. III) used to procure the measurements of the cross section (Sec. IV) and trans-verse single spin asymmetry (Sec. V). A final section is reserved for discussion of the results derived from these measurements.
II. EXPERIMENT A. RHIC polarized p þ p collider
The Relativistic Heavy Ion Collider (RHIC) is a particle accelerator located at Brookhaven National Laboratory. RHIC has the capability of bunching, storing, accelerating, and colliding polarized protons[45], as well as other ions, over a broad range of center-of-mass energies (pffiffiffis¼ 62.4 to 510 GeV for polarized protons). The injected beam into RHIC is typically made up of 111 bunches of polarized protons, which contain up toOð1011Þ protons per bunch for pþ p collisions and are collided at several different points around the ring. One such interaction point is located at the PHENIX experiment[46]. For the 2008 RHIC pþ p running, PHENIX (Fig. 1) consisted of two spectrometer arms at central pseudorapidityjηj < 0.35, two muon arms at pseudorapidity1.2 < jηj < 2.4, two global detectors, and two calorimeters (called the MPC detector) at forward pseudorapidity3.1 < jηj < 3.9.
A key aspect of the asymmetry measurements is the ability to align the spin vectors of the protons in the beam in a desired direction. The net fraction of protons in the beam with their spin vectors aligned along this desired direction is called the polarization (P). This must be measured to provide the correct scale for any asymmetry measurement. The polarization of the beams in RHIC is determined to within an uncertaintyΔP=P ∼ 4%–7% using two different kinds of polarimeters: a proton-carbon polarimeter[47]and a hydrogen-jet polarimeter[48]. The proton-carbon polar-imeter provides fast relative measurements of the polari-zation several times during a fill, while the hydrogen-jet polarimeter measurement takes several hours but yields the absolute polarization.
The polarization direction alternates for consecutive bunches which minimizes potential time-dependent and spin-dependent systematic uncertainties. In particular, detector efficiency and acceptance effects are minimized, as spin direction alternation in bunches allows use of the
same detector for both polarization directions. During the 2008 RHIC run, the average clockwise beam (also known as the blue beam) polarization was measured to be P¼ 0.490 0.021, while the average counterclockwise beam (yellow) polarization was P¼ 0.410 0.030. The stable polarization direction in RHIC is transverse, i.e., perpendicular to the accelerator plane.
B. PHENIX local polarimetry
The polarization direction is also measured locally at PHENIX using a pair of zero-degree calorimeters (ZDCs). The ZDCs comprise two hadronic calorimeters, located 18 m from the nominal PHENIX interaction point. A shower maximum detector (SMD) combined with the ZDC measures the transverse single spin asymmetry of very forward (η ≳ 6) neutrons which is found to be nonzero, and as large as AN∼10% [49,50]. A study of neutron AN in 2008 using the ZDC/SMD showed that the north-going (blue) polarization axis was oriented off-vertical by ϕblue¼ 0.263 0.03ðstatÞ 0.090ðsystÞ radians. The south-going polarization axis was found to be consistent with the nominal vertical direction, ϕyellow¼ 0.019 0.048ðstatÞ 0.103ðsystÞ.
West
South Side View
Beam View PHENIX Detector 2008 North East MuTr MuID MuID RxNP MPC RxNP PbSc PbSc PbSc PbSc PbSc PbGl PbSc PbGl TOF-E PC1 PC1 PC3 PC2 Central Magnet Central Magnet
North Muon Magnet South Muon Magnet
TEC PC3 BBC MPC BB RICH RICH DC DC ZDC North ZDC South Aerogel TOF-W 7.9 m = 26 ft 10.9 m = 36 ft 18.5 m = 60 ft
FIG. 1 (color online). The PHENIX detector configuration
C. PHENIX beam-beam counters
The beam-beam counters (BBC) (see Fig. 1) comprise two arrays of 64 quartz Čerenkov radiators connected to photomultiplier tubes (PMTs). The BBC is z¼ 144 cm from the nominal interaction point and covers 3.0 < jηj < 3.9. The primary functions of this detector are to measure the position of the collision along the beam (z) axis to a precision of σðzvertexÞ ¼ 2 cm, to provide a minimally biased trigger, and to measure the luminosity.
D. PHENIX MPC detector
The muon piston calorimeter (MPC) comprises two forward electromagnetic calorimeters, referred to as the south and north MPCs (see Fig.1), placed220 cm from the nominal interaction point along the beam axis. The south (north) MPC is made up of 196 (220) 2.2 × 2.2 × 18 cm3 PbWO
4 crystal towers and is read out with Hamamatsu S8664-55 avalanche photodiodes (APD). The MPC covers the pseudorapidity regions −3.7 < η < −3.1 and 3.1 < η < 3.9, respectively. The primary goal of the MPC is to identifyπ0andη mesons and measure their energy.
PbWO4 crystals were chosen for their short radiation length (0.89 cm) and small Molière radius (2.0 cm). Similar PbWO4crystals were originally used and extensively tested for the PHOS detector[51], part of the ALICE experiment at CERN. The MPC is not cooled and runs at the ambient temperature of its location in PHENIX. The gain variation with time, due largely to temperature variations and radiation damage to the crystals and APDs, is tracked using a LED calibration system. The absolute gain cali-bration comprises the LED tracking and tower by tower calibrations usingπ0s. The relative energy resolution after calibration was found to be σðEÞ=E ¼ 13%=pffiffiffiffiE⊕8%. Comparisons between theπ0and η mesons using real data and simulations showed that an overall energy scale uncer-tainty of 2% remained after all the calibrations, and also determined that the position resolution for clusters was about 2 mm. A schematic of the north MPC is given in Fig. 2.
E. Triggers
Readout of the PHENIX detector was done using one of two independent triggers for this analysis. The minimum bias (MB) trigger initiated readout when at least one BBC PMT in each array is hit, and when the collision vertex is within jzj < 30 cm of the nominal interaction point in PHENIX. As the number of collisions delivered by RHIC exceeds the data-taking rate of the PHENIX data acquisition system, only a small fraction of events can be recorded with “minimum bias.” To enhance the rarer (higher momentum) η mesons in the data stream an additional trigger is used to record the high-pT part of the cross section. This higher momentum trigger (called the 4 × 4B trigger) records an event when the total sum in any of the4 × 4 trigger arrays of
MPC towers satisfies an energy threshold of E≳ 20 GeV. The4 × 4 trigger arrays are particular groupings of towers and are called tiles. Each tile overlaps by two towers in the horizontal and vertical directions, as shown in Fig. 2, to provide even coverage for the trigger over the whole detector. The4 × 4B trigger is formed without the require-ment of a collision vertex from the BBCs.
III. IDENTIFICATION OFη MESONS IN THE MPC To identify η mesons in the MPC, the decay channel η → γγ is used which has a branching ratio of BR ¼ 0.3941 0.0020 [52]. Clusters of MPC towers from a single event are combined to form photon candidates. To increase the likelihood that a cluster is due to a real photon, clusters which do not possess the characteristic electro-magnetic shower shape are discarded. Clusters with their central tower tagged as noisy or inactive are also removed from the analysis. Once a sample of clusters is reduced to an enhanced sample of real photon candidates, clusters are paired together and an invariant mass is calculated, Eq.(1), Mγγ ¼pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4 · E1· E2· sinðθ12=2Þ; ð1Þ where E1;2 is the measured energy of each cluster, andθ12 is the opening angle between the momentum vectors of the two clusters. Additional kinematic cuts are made on paired clusters for the minimum bias and4 × 4B data sets. A minimum energy E1þ E2>7 GeV and 10 GeV, respectively, is imposed. A maximum energy asymmetry, α ¼ jE1−E2
E1þE2j, of α ¼ 0.6 and 0.8, respectively, is required.
The difference in the energy asymmetry cut between the
FIG. 2 (color online). A schematic of the north MPC as it appears
in PHENIX. The dotted [red and blue] squares drawn on the MPC
two triggers is due to differences in the signal to back-ground figure of merit. Finally, the separation between the two clustersΔR has to be greater than 2.6 cm, minimizing merging effects between cluster showers. After the appli-cation of these cuts, the invariant mass is calculated for all pairs, which is shown in Fig.3as open symbols.
IV. THE η MESON CROSS SECTION The cross section can be written in terms of measured quantities as Ed 3σ h dp3 ¼ 1 Lpp;inel 1 2πpT ΔNmeas η
BR ·ϵreco·ϵtrigΔpTΔy
; ð2Þ
whereΔNmeas
η is the number of measured (raw)η mesons over a rapidity rangeΔy and transverse momentum interval ΔpT. NoteΔy ≈ Δη for η mesons at forward rapidity at the pT measured in this analysis. The data are scaled by the integrated luminosity (Lpp;inel) and the branching fraction, BR, for this decay channel. To account for inefficiency in triggering and reconstruction, the ΔNmeas
η is corrected by factors ϵtrig and ϵreco, respectively. Each of these compo-nents is described in the following sections.
A. Integrated luminosity (Lpp;inel)
The luminosity is calculated as the ratio of the number of minimum bias events sampled for each trigger condition, withinjzj < 30 cm, divided by the part of the p þ p cross section to which the BBCs are sensitive. This cross section is σBBC
pp ¼ 23.0 2.3 mb which is determined using a Vernier scan procedure[53]. The total integrated luminos-ity of the minimum bias data set isLMB¼ 0.0192 pb−1and that of the 4 × 4B data set is L4×4B¼ 3.87 pb−1.
B. Yield extraction (ΔNmeas η )
The invariant mass distribution (Fig.3) has two distinct components: correlated pairs (for example from η meson
decays) and uncorrelated (combinatorial) background pairs, due to pairing of clusters from different parent sources. To account for this combinatorial background in the minimum bias data set (0.5 < pT <3.0 GeV=c), pho-ton candidates are analyzed from different events (which necessarily removes all real combinations) to form a mixed event distribution. The mixed event pair distribution is normalized (green closed circles in Fig.3) to the real pair distribution by taking the ratio of the real and mixed distributions and fitting with a constant at high invariant mass, and then subsequently scaling the mixed event distribution by this constant. The subtraction from this real pair distribution results in a final γγ invariant mass spectrum which has all uncorrelated background pairs removed (blue closed circles in Fig. 3). Using the same mixed event procedure, only a small fraction of the4 × 4B background was found to be uncorrelated, and the rest is made from a jet correlated background made primarily fromπ0decays. The mixed-event subtraction removes only a small fraction of the uncorrelated background in the 4 × 4B triggered data set, so it is not applied in this case [see Figs.3(c) and3(d)].
Raw yields are extracted by fitting the invariant mass distributions (mixed-event subtracted in minimum bias sample) with a function for the correlated background plus a constant term times a normalized Gaussian distribution representing the signal peak (gray lines in Fig. 3). The optimal background function for the minimum bias (4 × 4B) data set was an exponential (gamma distribution) function. Variation of the functional form of the back-ground (second, third order polynomial) was used to evaluate the systematic uncertainty on the yield extraction.
C. Efficiency corrections (ϵreco andϵtrig) Measured (raw) yields must be corrected for reconstruc-tion and trigger inefficiencies. Simulareconstruc-tions are used to calculate the reconstruction efficiency (ϵreco), which
] 2 Mass [GeV/c 0.4 0.5 0.6 0.7 Counts 0 2000 4000 6000
(a) Minimum Bias
<1.25 GeV/c
T
1.00<p
Real Pairs Mixed Event Pairs
Real-Mixed Pairs Fit ] 2 Mass [GeV/c 0.4 0.5 0.6 0.7 Counts 0 20 40 60 (b) Minimum Bias <2.75 GeV/c T 2.50<p ] 2 Mass [GeV/c 0.4 0.5 0.6 0.7 Counts 0 1000 2000 3000 (c) 4×4B Trigger <2.75 GeV/c T 2.50<p Real Pairs Fit ] 2 Mass [GeV/c 0.4 0.5 0.6 0.7 Counts 0 50 100 150 200 250 (d) 4×4B Trigger <3.75 GeV/c T 3.50<p
FIG. 3 (color online). The invariant mass distribution for minimum bias [panels (a) and (b)] and4 × 4B [panels (c) and (d)] samples. In
all panels, open red circles represent all real pairs formed from MPC clusters. In panels (a) and (b) the small green closed symbols show the combinatorial background from mixed events (see text) and the closed blue symbols show the combinatorial-subtracted real pairs.
Panels (b) and (c) show the same pTselection and illustrate the importance of triggering to enhance the statistical significance at large
corrects for geometric acceptance and detector resolution effects. To produce an η meson pT spectrum which is similar to that in real data, a full Monte Carlo sample of single η mesons are initially generated flat in pT and pseudorapidity in the MPC kinematics, and with the same z-vertex distribution as measured in data. These generated single η mesons are passed through a GEANT (3.21) [54] description of the PHENIX detector and subsequent energy deposits are embedded into real data minimum bias events. Minimum bias events here do not necessarily contain anη meson from the collision. The same cluster identifica-tion and pair cuts are applied, followed by the full reconstruction, similar to that in the real data analysis.
The next step weights the reconstructed and generatedη mesons in pT and pseudorapidity to mimic the measured data distribution. This accounts for pTsmearing effects on an exponential spectrum, and for the falling pseudora-pidity dependence in the forward region. As the weighting is dependent on the shape of the corrected spectrum, an iterative procedure is used to ensure the efficiency correction converges to a stable value. The reconstruc-tion efficiency is calculated as the ratio of reconstructedη mesons divided by the number generated. The reconstruction efficiency for the south and north MPC for both triggers is shown in Fig.4. The north MPC has a lower reconstruction efficiency than the south, due to a more restrictive noisy/inactive tower map in the north. The reconstruction efficiency shape is predominantly due to the geometric acceptance coupled to the narrowing γγ opening angle from low to higher momenta. At low momenta, wider opening angles can prohibit the meas-urement of both γs in the detector. At high momenta, cluster merging increasingly inhibits the detection of distinctγ pairs. Significant cluster merging effects occur
when the cluster separation is less than 1.5 times the tower width (ΔR < 3.3 cm).
The trigger efficiency (ϵtrig) is estimated by taking the ratio ofη meson yields found using the trigger of interest (for example minimum bias) in coincidence with any other trigger which is unrelated (unbiased) divided by the same unrelated trigger without the coincidence requirement,
ϵηtrig¼
Nηunbias∧trig
Nηunbias : ð3Þ
For the minimum bias trigger efficiency,ϵηMB, the4 × 4B trigger is used as this maximizes the η meson yield statistics. The measured minimum bias trigger efficiency is found to beϵηMB¼ 0.76 0.01ðstatÞ 0.06ðsystÞ. There is a slight dependence on pT, which has been factored into the systematic uncertainty.
For the4 × 4B trigger efficiency for η mesons, ϵη4×4B, the minimum bias trigger is used as the unrelated trigger. The statistics in the minimum bias sample is limited, however, and the efficiency can only be determined from the data up to pT <3.0 GeV=c [see Fig.5(c), open symbols]. Instead, the trigger efficiency forη mesons in the 4 × 4B triggered sample is calculated by simulating the4 × 4B trigger.
The 4 × 4B trigger comprises a total of 56 (61) over-lapping 4 × 4 tower array sums from the south (north) MPC. An example of the efficiency of an individual4 × 4 array from data is shown in Fig.5(a). This efficiency is fit with a double error function
fðxÞ ¼ Z x
−∞½ag1ðx
0Þ þ ð1 − aÞg
2ðx0Þdx0; ð4Þ where g1ðxÞ and g2ðxÞ are Gaussian distributions. The efficiency curve shown in Fig.5(a)covers the entire data-taking period, and relative gain changes throughout the RHIC run due to temperature variations and radiation damage to the detector cause a large spread in the rise of the efficiency curve. This gain variation is monitored with an LED calibration system. The trigger threshold (θthresh) at any given instant is a step function and is thus implemented in the simulation as a step function. The changes in the effective threshold due to the gain variation over the run are accounted for in the simulation by varying the threshold using the data from the LED monitoring. Fit parameters from the 117 different trigger tile efficiency curves are derived and used in the trigger simulation to determine an optimalθthresh, and thus trigger efficiency for η mesons.
To tune the trigger simulation, reconstructed pþ p events fromPYTHIA(tune A)[55]were processed through the trigger simulation and matched to real data. The cluster trigger efficiency is well reproduced in the simulation when using a mean tile trigger threshold of θthresh¼ 0.66, determined from the fit parameters in Eq. (4) [see solid
[GeV/c] T Transverse Momentum, p 0 2 4 6 Ef ficiency [%] 0 5 10 South MPC Minimum Bias 4B Trigger × 4 North MPC Minimum Bias 4B Trigger × 4
FIG. 4 (color online). Reconstruction efficiency forη mesons
using the minimum bias (4 × 4B) data set, shown as open
(closed) symbols. The red circle (blue square) symbols show
line in Figs.5(a)and5(b)]. On average,θthreshcorresponds tohE4×4i ≈ 40 GeV. The comparison is shown in Fig.5(b), where good agreement is seen between the simulation of the 4 × 4B trigger and the data efficiency curve to all energies of interest in this analysis. Variations of the threshold in the simulation between 0.60 < θthresh<0.75 [see dotted (0.60) and dashed (0.75) lines in Figs. 5(a)
and5(b)] are used to estimate a systematic uncertainty on reproducing the 4 × 4B cluster trigger efficiency. These systematic variations account for differences in the south and north MPC, and for a turn-on uncertainty which occurs for low energy clusters that are smeared out above and below the selected trigger turn-on.
Within this trigger simulation framework, the 4 × 4B trigger efficiency forη mesons is calculated from the same single-η simulations used in the reconstruction efficiency study. This simulation accounts for effects such as when the distance between the two decay photons, ΔR, is small enough that the two photons fall into the same 4 × 4 tile such that their energy sum fires the trigger together. Figure 5(c) shows the η meson 4 × 4B trigger efficiency calculated via simulation, with a comparison to the sta-tistically limited values measured from the minimum bias trigger in the overlap region of 2.0 < pT <3.0 GeV=c calculated using Eq.(3). In this overlap region there is good agreement within statistical (shown) and systematic uncer-tainties (not shown; see next section).
D. Systematic uncertainties
The systematic uncertainties are divided into three types. Statistical and point-to-point uncorrelated systematic uncertainties are added in quadrature to form type-A uncertainties. Type B represents correlated uncertainties between pT bins. Type C is external global systematic uncertainties which underlay the measurement.
The functional form of the background used in the yield extraction was varied and contributes 5%–15% to the
type-A uncertainties. The systematic uncertainty due to energy scale (type B) was found to vary from 3% to 30% for pT ¼ 0.5 to 5.0 GeV=c. A global reconstruction efficiency uncertainty (type B) of 11.5% (27.5%) is applied for pT >0.75 GeV=c (pT <0.75 GeV=c). An additional reconstruction efficiency uncertainty of 1% to 20% for pT ¼ 3.0 to 5.0 GeV=c is assigned due to cluster merging effects (type B). The systematic uncertainty on varying the turn-on threshold (type B) for the 4 × 4B trigger efficiency leads to 30% uncertainty at pT ¼ 2.0 GeV=c, which decreases exponentially to 5% at pT ¼ 5.0 GeV=c. A further global (type C) systematic uncertainty of 9.7% is applied based on the luminosity monitoring of the BBC.
E. Cross section results
The cross section is calculated using Eq.(2) independ-ently for the south and north MPC, and for both the minimum bias and 4 × 4B data sets. For both trigger conditions, the south and north reconstructed cross sections agree to within 2% across pT. The south and north cross sections measured for each trigger are weighted together to determine the final cross section spectrum. Agreement in the overlap region (2 < pT <3 GeV=c) between the mini-mum bias and4 × 4B cross section was within 7% across pT and is within the type-A systematic uncertainties. In the overlapping region, data points from the two data sets are combined as a weighted average.
The invariant cross section of η mesons is shown in Fig.6and TableIas a function of transverse momentum, measured between0.5 < pT <5.0 GeV=c within a pseu-dorapidity range of3.0 < jηj < 3.8. The results are com-pared to a next-to-leading-order (NLO) pQCD calculation for three different choices of scaleμ[56,57], over the same pseudorapidity region as the measurement. Here,μ repre-sents the factorization, renormalization, and fragmentation scales, which are set to be equal to one another.
Ef ficiency [%] 0 50 100 ADC [a.u.] 1000 2000 3000
Cluster Energy [GeV]
20 40 [GeV/c] T Transverse momentum, p 2 3 4 5 (a)
Trigger Tile Efficiency Fit (b) Data Simulated (c) South MPC Data Simulation North MPC Data Simulation
FIG. 5 (color online). (a) The trigger efficiency for a single4 × 4 tower array in the 4 × 4B trigger. The solid, dashed, and dotted lines
representθthresh¼ 0.66, 0.60, and 0.75, respectively. (b) A comparison of the cluster efficiency as a function of energy to the simulated
efficiency generated using the differentθthresh. (c) Theη meson 4 × 4B trigger efficiency, ϵη4×4B(systematic error not included). The open
symbols representϵη4×4Bcalculated using Eq.(3)with the minimum bias trigger as the unrelated trigger. The closed points represent
The lower panel shows the comparison between the measured cross section and the NLO pQCD. For pT >2.0 GeV=c, the NLO pQCD calculation is in very good agreement with the measured cross section. Upon approaching the pQCD limit at low momentum (pT < 2.0 GeV=c) the agreement is less clear, but well within the factorization uncertainty.
V. THE TRANSVERSE SINGLE SPIN
ASYMMETRY FORη MESONS
In polarized p↑+p collisions, the cross section of hadron production can be modified in azimuth, with respect to the polarization direction. To first order the azimuthally dependent cross section can be written as
dσ dΩ¼ dσ dΩ 0ð1 þ Py · AN· cosϕÞ; ð5Þ whereðdΩdσÞ0is the unpolarized differential cross section, Py is the vertical beam polarization, and AN is the transverse single spin asymmetry. This dependence can be measured as
Py· AN· cosϕ ¼ ϵNðϕÞ; ð6Þ where ϵNðϕÞ is the measured raw asymmetry which, to first order, is an azimuthal cosine modulation. For this analysis, AN is found by first measuring the raw asym-metry [ϵNðϕÞ], fitting it with a cosine function, and then dividing the amplitude by the average beam polarization. The raw asymmetry is measured in this analysis using two methods[58].
The first method is known as the polarization formula, ϵpol
N ðϕÞ ¼
N↑ðϕÞ − N↓ðϕÞ
N↑ðϕÞ þ N↓ðϕÞ; ð7Þ which uses two different polarization yields (up—↑ and down—↓) in one azimuthal region. This method is pre-ferred if the acceptance is not homogeneous, but relative luminosity effects (R ¼L↑
L↓) must be taken into account.
A second method is known as the square-root formula, Eq. (8), which uses the geometric mean of the yields N from two azimuthal regions on opposite sides of the MPC (ϕ and ϕ þ π) and two polarization directions (up—↑ and down—↓). When there is little loss of acceptance, par-ticularly dead areas in azimuthal space, this method is advantageous as detector efficiency and relative luminosity effects cancel. ϵ ffiffiffi ϕ p N ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi N↑ðϕÞ · N↓ðϕ þ πÞ p −pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiN↓ðϕÞ · N↑ðϕ þ πÞ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi N↑ðϕÞ · N↓ðϕ þ πÞ p þpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiN↓ðϕÞ · N↑ðϕ þ πÞ: ð8Þ -9 10 -7 10 -5 10 -3 10 -1 10 10 2 10 -1 0 1 0 1 2 3 4 5 [GeV/c] T Transverse Momentum, p ] 3c -2 [mb GeV3 dp σ 3 d E (Data-pQCD)/pQCD = 200 GeV) s +X ( η → p+p <3.8 η 3.0< PHENIX (2008 data)
Global Scale Uncert. 9.7%
NLO pQCD /2 T = p μ T = p μ T = 2p μ
FIG. 6 (color online). The cross section of inclusiveη mesons
produced from pþ p collisions at pffiffiffis¼ 200 GeV at forward
rapidity. The upper panel shows the measured cross section
versus transverse momentum (pT), compared to an NLO pQCD
calculation at three different scales μ [56,57]. The lower panel
shows the difference between the measured cross section and each of the NLO pQCD calculations. Error bars (bands) represent type-A (type-B) systematic uncertainties. A global scale uncer-tainty (type-C, 9.7%) is due to the luminosity and global reconstruction uncertainties.
TABLE I. The measuredη meson cross section versus pT at
forward rapidity for the 2008 data set with statistical and systematic (type-A and type-B) uncertainties. There is an addi-tional normalization uncertainty of 9.7% (type C).
pT ½GeV=c Ed
3σ
dp3 [mb GeV−2c3] Type A Type B
0.625 6.03 × 10−1 8.76 × 10−2 1.68 × 10−1 0.875 1.80 × 10−1 3.12 × 10−2 2.61 × 10−2 1.125 6.39 × 10−2 4.48 × 10−3 9.71 × 10−3 1.375 2.15 × 10−2 8.17 × 10−4 3.35 × 10−3 1.625 7.61 × 10−3 3.98 × 10−4 1.17 × 10−3 1.875 2.61 × 10−3 1.31 × 10−4 4.08 × 10−4 2.125 1.07 × 10−3 5.31 × 10−5 1.59 × 10−4 2.375 4.35 × 10−4 2.04 × 10−5 6.33 × 10−5 2.625 1.72 × 10−4 6.39 × 10−6 2.39 × 10−5 2.875 7.68 × 10−5 3.08 × 10−6 1.13 × 10−5 3.125 3.42 × 10−5 1.19 × 10−6 8.42 × 10−6 3.375 1.43 × 10−5 8.87 × 10−7 3.53 × 10−6 3.625 6.61 × 10−6 5.96 × 10−7 1.62 × 10−6 3.875 3.20 × 10−6 3.71 × 10−7 9.41 × 10−7 4.125 1.31 × 10−6 1.42 × 10−7 3.95 × 10−7 4.375 6.17 × 10−7 1.30 × 10−7 2.17 × 10−7 4.750 2.51 × 10−7 2.92 × 10−8 1.01 × 10−7
The final transverse single spin asymmetry result reported uses the square-root formula. The polarization formula serves as a cross-check.
A. Polarization
To measure AN, the polarization and spin information of only one beam is used, while the other beam’s spin information is ignored, such that it is integrated over to a net polarization of zero. As one chooses which beam to use as“polarized,” two independent AN measurements can be made: one utilizing the north-going beam’s polarization, and one utilizing the south-going beam’s polarization. Effectively, as the south and north MPC detectors are independent with differing systematics, two independent measures of AN are derived, allowing for more reliable evaluation of systematic uncertainties on the results.
B. AN analysis
To measure the raw AN, the ϕ distribution of the reconstructed η meson is divided into 12 azimuthal bins,
and spin dependent η meson yields are obtained for each bin.
To extract theη meson yields for the ANmeasurements, the invariant mass spectra from all photon pairs are first formed independent of spin direction andϕ, binned in xF (or pT). These invariant mass spectra are then fit with a signal Gaussian and background function. The signal Gaussian establishes the peak mass (Mη) and width (ση) which are used to define anη mass window for the given xF (pT) bin. The counts from the background function and signal are also used to form a relative contribution under the peak region from the background (r¼ NBG
NBGþNη).
Spin dependent andϕ dependent invariant mass spectra are then formed, with the spin and ϕ dependent yields determined by integrating the invariant mass spectra between Mη 2ση. An example of the signal and back-ground regions are shown in Fig.7.
The asymmetry in the peak regionϵM2σN is then simply calculated from Eqs.(7)and(8). The resultant asymmetries are then fit with a cosine function; see Fig. 8 using the square-root formula, Eq.(8). Note that Fig.8has six points, because azimuthal bins on opposite sides of the MPC are folded into each other when using the square-root formula.
] 2 Mass [GeV/c 0.4 0.6 0.8 Counts 0 2000 4000 6000 8000 <0.5 F 0.4<x
FIG. 7 (color online). Invariant mass spectrum for the south
MPC, illustrating theη meson peak region (solid fill), as well as
the sideband regions (diagonal fill and crosshatch).
[rad] φ -3 -2 -1 0 N ∈ -0.05 0 0.05 0.4<xF<0.5
FIG. 8 (color online). An example of a raw (square-root
method) asymmetry fit for a single xF bin in the south MPC.
TABLE II. AM2σN and A
bg
N forη mesons measured as a function
of xFfrom the4 × 4B triggered data set. The values represented
are the weighted mean of the south and north MPC. The uncertainties listed are statistical only.
xFbin AMN2σ Statistical A bg N Statistical −0.7 to −0.6 −0.0385 0.0602 0.0366 0.1256 −0.6 to −0.5 0.0110 0.0186 −0.0484 0.0360 −0.5 to −0.4 0.0094 0.0094 −0.0261 0.0178 −0.4 to −0.3 0.0135 0.0117 0.0186 0.0199 0.3 to 0.4 0.0314 0.0127 0.0028 0.0208 0.4 to 0.5 0.0537 0.0102 0.0242 0.0190 0.5 to 0.6 0.0353 0.0196 0.0458 0.0380 0.6 to 0.7 0.0974 0.0628 0.0147 0.131 F x -0.5 0 0.5 N A -0.2 0 0.2 =200GeV) s +X ( η → ↑ p+p
Vertical scale uncert. 4.8%
< 0 F x > 0 F x
FIG. 9 (color online). The xF dependence of AN. The vertical
error bars show the statistical uncertainty, the blue bands represent uncorrelated systematic uncertainties (see text for details). The relative luminosity effect systematic uncertainties
The transverse single spin asymmetry in the η meson peak region is then calculated, AM2σN , using Eq. (6). As mentioned, the amplitude of the cosine function, divided by the beam polarization, gives the value of AM2σN . Because a significant background remains under the η mass region, the final measurement of AN must be corrected for any dilution of the asymmetry due to this background. This background is composed of non-η meson particles, which may have a different asymmetry than the signalη mesons. The correction is obtained from the asymmetry measured from a combined mass region from regions below (M−5σ < minv< M−3σ GeV=c2) and above (M3σ< minv< M5σ GeV=c2) the η meson mass peak, shown as the diagonal and crosshatch filled regions in Fig. 7, respec-tively. The background-corrected η meson asymmetry is expressed as AηN¼A M2σ N − rA bg N 1 − r ; ð9Þ
where r is the background fraction in the2σ region around theη mass peak, AM2σN is the measured asymmetry of the peak region, and AbgN is the measured asymmetry of the background regions. The r values are found from the spin-independent signal and background invariant mass spectrum fits mentioned above. For the lowest xFbins, calculated from the minimum bias data, hrMBi ¼ 0.60. For the highest xF bins, calculated from the4 × 4B data, hr4×4Bi ¼ 0.37. AbgN was found to be consistent in the low and high mass regions. Overall the background correction from Eq. (9) had a moderate effect of AηN > AM2σN . Table II summarizes AM2σN and AbgN from the4 × 4B triggered data set.
C. AN Results
The xF-dependent AN is shown in Fig.9and TableIII, based on the weighted mean of the measured south and north MPC AηN values. The average pseudorapidity of the
measuredη mesons is hηi ¼ 3.52. The procedure to obtain ANfrom the minimum bias triggered data set is the same as that in the4 × 4B data set, and where the triggers overlap in xF, the AN values are weighted together. For forward xF (xF >0), a clear rising asymmetry is seen, ranging from 2% to 20% over the measured xF range. For backward xF (xF <0), ANis flat and consistent with zero when averaged
TABLE III. ANforη mesons measured as a function of xF. Uncertainties listed are those due to the statistics, the
xFuncorrelated uncertainties due to extracting the yields, and the correlated relative luminosity uncertainty (see text
for details).
Uncertainty
xF bin hxFi hpTi [GeV=c] AηN Statistical Uncorrelated Correlated
−0.7 to −0.6 −0.63 3.41 −0.0503 0.1054 0.0791 0.0024 −0.6 to −0.5 −0.535 3.04 0.0417 0.0319 0.0385 0.0023 −0.5 to −0.4 −0.444 2.68 0.0376 0.0165 0.0161 0.0021 −0.4 to −0.3 −0.358 2.34 0.0094 0.0219 0.0095 0.0023 −0.3 to −0.2 −0.231 1.35 0.0226 0.0339 0.0179 0.0000 0.2 to 0.3 0.231 1.35 0.0212 0.0342 0.0204 0.0000 0.3 to 0.4 0.358 2.34 0.0491 0.0232 0.0127 0.0020 0.4 to 0.5 0.444 2.68 0.0792 0.0177 0.0083 0.0018 0.5 to 0.6 0.535 3.04 0.0372 0.0335 0.0179 0.0020 0.6 to 0.7 0.629 3.41 0.1939 0.1092 0.0392 0.0019 F x 0.2 0.4 0.6 0.8 N A 0.0 0.1
0.2 PHENIX ηπ (0 (3.1<〈η〉=3.52,η <3.7,s=200GeV)s=62.4GeV) STAR (〈η〉=3.3, s=200GeV) 0 π =200GeV) s =3.7, 〉 η 〈 ( 0 π E704 π0 (1.0<η<4.6,s=19.4GeV) (a) F x 0.2 0.4 0.6 0.8 N A 0.0 0.2 0.4
KK Twist-3 η (3.1<η<3.7, 1<pT<5GeV/c, s=200 GeV) PHENIX η (〈η〉=3.52, s=200GeV)
STAR η (〈η〉=3.68, s=200GeV) E704 η (1.0<η<4.6, s=19.4GeV)
(b)
FIG. 10 (color online). Comparison between theη meson AN
and other results. Panel (a) compares withπ0 meson AN results
from PHENIX[15], STAR[13], and E704[9]in red circle, blue
star, and green square symbols, respectively. Panel (b) compares
to the STAR η meson AN result [14] (blue stars), the E704 η
meson AN result[11](green squares), and a twist-3 calculation
over xF within 1.7σ of the statistical plus systematic uncertainties. An uncorrelated systematic uncertainty is shown as bands around the points and is found by varying the functional form of the background functions. This changes the M 2σ range and relative r values, which affects the number ofη mesons used in the calculation of AN. It also includes systematic uncertainty estimation from three different cross-checks on the measurement of AN: increasing the mass window to M 2.5σ, the difference from the polarization formula measurement [Eq.(7)], and adding higher order cosine terms to the raw asymmetry fit. The correlated systematic uncertainty (not shown in Fig.9;
see TableIII) is due to small residual relative luminosity effects in the square-root formula.
Figure10shows the measured ANforη mesons compared to other AN measurements. The upper panel shows a comparison between η meson and π0 meson asymmetries in overlapping xF and similar pseudorapidity ranges at various collision energies. The η meson AN is similar to theπ0 AN measurements at a lower center-of-mass energy made by the PHENIX experiment using the MPC[15], as well asπ0 from the E704[9]and STAR[13]experiments. The similarity between theη and π0 asymmetries suggests that initial-state spin-momentum correlations could play a role, or a common spin-momentum correlation is present in the fragmentation ofπ0 andη mesons.
The lower panel of Fig. 10 shows a comparison to measurements made by E704[11] (pffiffiffis¼ 19.4 GeV) and STAR[14]at the same collision energy (pffiffiffis¼ 200 GeV). The average pseudorapidity of the PHENIX result is hηi ¼ 3.52, while the average pseudorapidity of the STAR result is hηi ¼ 3.68. For xF >0.55, the STAR η meson AN is larger than this PHENIX η meson AN measurement, but these two results are consistent with each other within type-A uncertainties.
The asymmetries in Fig. 10 are compared to a twist-3 calculation by Kanazawa and Koike [59] based on [40], performed for the PHENIX kinematics. It describes the magnitude of the asymmetry well at the lowest and highest points in xF, but it is unclear whether the observed shape for the middle xF values is well described. No theoretical uncertainty on the calculation is available at this time; a better understanding of the theoretical uncertainties will be
[GeV/c] T Transverse momentum, p 1 2 3 4 N A 0 0.1 0.2
0.3 p+pVertical scale uncert. 4.8%↑→η+X (s=200GeV)
< 0 F x > 0 F x <3.7) η > 0.2, 3.1< F (x η KK Twist-3
FIG. 11 (color online). The pTdependence of AN. The vertical
error bars show the statistical uncertainty; the blue bands represent uncorrelated systematic uncertainties (see text for details). The relative luminosity effect systematic uncertainties
are not shown (see text and TableIV). The purple line shows a
prediction from a twist-3 calculation based on quark-gluon
correlation functions[59].
TABLE IV. ANforη mesons measured as a function of pT. Uncertainties listed are those due to the statistics, the
pT uncorrelated uncertainties due to extracting AN, and the correlated relative luminosity uncertainty (see text for
details).
Uncertainty
pT bin [GeV=c] hpTi [GeV=c] hxFi AηN Statistical Uncorrelated Correlated
xF<−0.2 1.0 to 1.5 1.24 0.23 0.0370 0.0401 0.0117 0.0000 1.5 to 2.0 1.68 0.27 0.0189 0.0512 0.0233 0.0000 2.0 to 2.5 2.27 0.42 0.0355 0.0228 0.0183 0.0042 2.5 to 3.0 2.73 0.44 0.0343 0.0191 0.0136 0.0041 3.0 to 3.5 3.21 0.46 0.0214 0.0259 0.0149 0.0047 3.5 to 4.0 3.70 0.48 −0.0147 0.0452 0.0213 0.0053 4.0 to 4.5 4.19 0.51 0.0211 0.0887 0.0822 0.0057 xF>0.2 1.0 to 1.5 1.24 0.23 0.0143 0.0409 0.0131 0.0000 1.5 to 2.0 1.68 0.27 0.0511 0.0514 0.0120 0.0000 2.0 to 2.5 2.27 0.42 0.0713 0.0251 0.0176 0.0042 2.5 to 3.0 2.73 0.44 0.0605 0.0206 0.0085 0.0041 3.0 to 3.5 3.21 0.46 0.0564 0.0274 0.0078 0.0047 3.5 to 4.0 3.70 0.48 0.1443 0.0480 0.0306 0.0053 4.0 to 4.5 4.19 0.51 0.1066 0.0944 0.0257 0.0057
necessary in order to draw a quantitative conclusion on the agreement with data.
The pTdependence of the asymmetry is shown in Fig.11 and TableIV. For ANmeasured at forward xF (xF >0.2), a clear nonzero asymmetry is seen (hANi ¼ 0.061 0.012), while AN for backward xF (xF <−0.2) is consistent with zero within1.7σ. The uncorrelated and correlated system-atic uncertainties are evaluated the same way as in the xF dependence of AN.
Figure11also shows the measured AN as a function of pTcompared to the twist-3 calculations. Similar to the case for the xFdependence, the twist-3 calculation describes the magnitude of the asymmetry well at the lowest and highest measured points in pT, but it is not clear if it describes the observed shape in the mid-pTrange. It should be noted that the data points in pT are integrated over a wide range of xF,0.2 < xF <0.7.
VI. SUMMARY
By utilizing data taken by the MPC detector installed at forward rapidity in the PHENIX experiment at RHIC, the invariant cross section as a function of pT and the transverse single spin asymmetry AN as a function of xF and pT have been measured for inclusive η mesons produced at forward rapidity (hηi ¼ 3.52) from p↑þ p collisions at a center-of-mass energy of pffiffiffis¼ 200 GeV. The NLO pQCD calculation was found to be consistent with the invariant cross section measurement at momentum of pT >1.5 GeV=c. This measurement can be used to improve constraints on the hadronization process of η mesons in future global analyses of the η fragmentation function. Nonzero asymmetries measured at forward xFare consistent with previousπ0meson results within statistical uncertainties. Because the π0 and η mesons are produced from potentially different parton fractions, and also might have different polarized fragmentation functions due to isospin or mass differences or the presence of strange quarks in theη, these data will help to constrain the relative importance of spin-momentum correlations in the initial-state polarized protons versus that of spin-momentum correlations in the fragmentation. The dependencies of the measured asymmetry on xFand pT are reasonably well described by twist-3 calculations using quark-gluon corre-lation functions; a quantitative comparison can be made once uncertainties become available on the calculations.
With higher statistics from future data sets, a doubly differential measurement of the asymmetry binned in both xF and pT simultaneously could provide a much more stringent test of any available calculations and better constrain twist-3 quark-gluon correlation functions if they turn out to be the dominant contribution.
ACKNOWLEDGMENTS
We thank the staff of the Collider-Accelerator and Physics Departments at Brookhaven National Laboratory and the staff of the other PHENIX participating institutions for their vital contributions. We acknowledge support from the Office of Nuclear Physics in the Office of Science of the Department of Energy, the National Science Foundation, Abilene Christian University Research Council, Research Foundation of SUNY, and Dean of the College of Arts and Sciences, Vanderbilt University (USA), Ministry of Education, Culture, Sports, Science, and Technology and the Japan Society for the Promotion of Science (Japan), Conselho Nacional de Desenvolvimento Científico e Tecnológico and Fundação de Amparo à Pesquisa do Estado de São Paulo (Brazil), Natural Science Foundation of China (P. R. China), Ministry of Science, Education, and Sports (Croatia), Ministry of Education, Youth and Sports (Czech Republic), Centre National de la Recherche Scientifique, Commissariat à l’Énergie Atomique, and Institut National de Physique Nucléaire et de Physique des Particules (France), Bundesministerium für Bildung und Forschung, Deutscher Akademischer Austausch Dienst, and Alexander von Humboldt Stiftung (Germany), OTKA NK 101 428 grant and the Ch. Simonyi Fund (Hungary), Department of Atomic Energy and Department of Science and Technology (India), Israel Science Foundation (Israel), National Research Foundation of Korea of the Ministry of Science, ICT, and Future Planning (Korea), Physics Department, Lahore University of Management Sciences (Pakistan), Ministry of Education and Science, Russian Academy of Sciences, Federal Agency of Atomic Energy (Russia), VR and Wallenberg Foundation (Sweden), the U.S. Civilian Research and Development Foundation for the Independent States of the Former Soviet Union, the Hungarian American Enterprise Scholarship Fund, and the U.S.-Israel Binational Science Foundation.
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![FIG. 3 (color online). The invariant mass distribution for minimum bias [panels (a) and (b)] and 4 × 4B [panels (c) and (d)] samples](https://thumb-ap.123doks.com/thumbv2/123dokinfo/5080724.74367/7.918.118.802.776.980/color-online-invariant-distribution-minimum-panels-panels-samples.webp)
